An object placed at a distance of a 9cm from the first principal focus of a convex lens produces a real image at a distance of 25cm. from its second principal focus. then the focal length of the lens is :

Options

A. 9 cm
B. 25 cm
C. 15 cm
D. 17 cm

Answer

To find the focal length of the convex lens based on the given information, we can follow these steps:

### Step 1: Understand the problem
We know that an object is placed at a distance of 9 cm from the first principal focus (F1) of the lens, and it produces a real image at a distance of 25 cm from the second principal focus (F2).

### Step 2: Set up the distances
Let the focal length of the lens be \( f \). The distance from the lens to the object (denoted as \( u \)) is given as:
\[ u = - (f + 9) \]
(The negative sign is used because the object is placed on the same side as the incoming light.)

The distance from the lens to the image (denoted as \( v \)) is given as:
\[ v = f + 25 \]
(The positive sign is used because the image is real and formed on the opposite side of the lens.)

### Step 3: Use the lens formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]

Substituting the values of \( u \) and \( v \) into the lens formula:
\[ \frac{1}{f} = \frac{1}{(f + 25)} - \frac{1}{(- (f + 9))} \]

### Step 4: Simplify the equation
This can be rewritten as:
\[ \frac{1}{f} = \frac{1}{(f + 25)} + \frac{1}{(f + 9)} \]

Now, finding a common denominator:
\[ \frac{1}{f} = \frac{(f + 9) + (f + 25)}{(f + 25)(f + 9)} \]
\[ \frac{1}{f} = \frac{2f + 34}{(f + 25)(f + 9)} \]

### Step 5: Cross-multiply
Cross-multiplying gives:
\[ (f + 25)(f + 9) = f(2f + 34) \]

### Step 6: Expand both sides
Expanding both sides:
\[ f^2 + 34f + 225 = 2f^2 + 34f \]

### Step 7: Rearrange the equation
Rearranging gives:
\[ 0 = 2f^2 - f^2 + 34f - 34f + 225 \]
\[ f^2 - 225 = 0 \]

### Step 8: Solve for \( f \)
This can be factored as:
\[ (f - 15)(f + 15) = 0 \]
Thus, \( f = 15 \) cm or \( f = -15 \) cm. Since we are looking for the focal length of a convex lens, we take the positive value:
\[ f = 15 \text{ cm} \]

### Final Answer
The focal length of the lens is \( 15 \text{ cm} \).

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